Abstract:Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the fundamental group $\pi _1(X)$ on the universal covering $\tilde {X}$ is of the first kind. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pants decomposition of $X$ we define a family of train tracks on $X$ such that any geodesic lamination on $X$ is weakly carried by at least one train track. The set of measured laminations on $X$ carried by a train track is in a one to one correspondence with the set of edge weight systems on the train track. Furthermore, the above correspondence is a homeomorphism when we equipped the measured laminations (weakly carried by a train track) with the weak* topology and the edge weight systems with the topology of pointwise (weak) convergence. The space $ML_b(X)$ of bounded measured laminations appears prominently when studying the Teichmüller space $T(X)$ of $X$. If $X$ has a bounded pants decomposition, a measured lamination on $X$ weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The space $ML_b(X)$ is equipped with the uniform weak* topology. The correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism for the uniform weak* topology on $ML_b(X)$ and the topology induced by supremum norm on the edge weight system.

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中文翻译：

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无限表面上的火车轨道和测量的叠片

摘要：设 $X$ 是一个无限黎曼曲面，其配备了共形双曲度规，使得基本群 $\pi_1(X)$ 对通用覆盖 $\tilde {X}$ 的作用是第一类。我们首先证明 $X$ 上的任何测地线叠层都不密集。给定 $X$ 的固定测地线裤分解，我们在 $X$ 上定义一系列火车轨道，使得 $X$ 上的任何测地线叠层都被至少一个火车轨道弱承载。由火车轨道承载的 $X$ 上的一组测量叠片与火车轨道上的一组边缘权重系统一一对应。此外，当我们为测量的叠层（由火车轨道弱承载）配备弱*拓扑和具有点（弱）收敛拓扑的边缘权重系统时，上述对应关系是同胚。在研究 $X$ 的 Teichmüller 空间 $T(X)$ 时，有界测量叠片的空间 $ML_b(X)$ 显得尤为突出。如果 $X$ 具有有界裤子分解，则当且仅当相应的边权重系统具有有限的最高范数时，由火车轨道弱承载的 $X$ 上的测量层压是有界的。空间 $ML_b(X)$ 配备了统一的弱*拓扑。由火车轨道弱承载的有界测量叠片与其边缘权重系统之间的对应关系是 $ML_b(X)$ 上的均匀弱*拓扑和边缘权重系统上由最高范数引起的拓扑的同胚。当且仅当相应的边权重系统具有有限的最高范数时，由火车轨道弱承载的 $X$ 上的测量层压是有界的。空间 $ML_b(X)$ 配备了统一的弱*拓扑。由火车轨道弱承载的有界测量叠片与其边缘权重系统之间的对应关系是 $ML_b(X)$ 上的均匀弱*拓扑和边缘权重系统上由最高范数引起的拓扑的同胚。当且仅当相应的边权重系统具有有限的最高范数时，由火车轨道弱承载的 $X$ 上的测量层压是有界的。空间 $ML_b(X)$ 配备了统一的弱*拓扑。由火车轨道弱承载的有界测量叠片与其边缘权重系统之间的对应关系是 $ML_b(X)$ 上的均匀弱*拓扑和边缘权重系统上由最高范数引起的拓扑的同胚。

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